image image | text string | source_file string | page_number int64 |
|---|---|---|---|
ααααα½αα’αααα αα»ααααα·αααΈα‘αΆ
# ααα·ααα·ααααΆ
## ααααα·αααααα
## ααααΆααααΈ α‘α‘
ααααα»αααααααΆαααα
ααααΉαααααΆαααααα»ααααα·αα
ααααααΆα
α’ααΆα α‘α€α¨ αα αΆαα·ααΈ ααααααααααα ααααααα | [11] Math - High | 1 | |
| | |
|---|---|
| **αααααααααΆααα·αααα** | |
| ααα α’αα»α αααΆαααΈ | αααααααΈ ααΈ αααΌααΈαααα |
| ααα α±α ααΈαααΆ | αααααααΈ α’αα»α αα»αααΈ |
| ααα α
αΆαα αααΆααΆ | ααα ααα»α αα½ |
| ααα ααΌ αααα | |
| **α’αααααΆαα’ααααα** | αααααααΈ ααΆα ααΆααΈα |
| **αα·α
α·ααααα** | ααα ααα ααΆαα· |
| **α’ααααααααα** | ααα α‘α»α αα»ααα | ααα αααα αα½α |
| **α’ααααα
ααΆααααα** ... | [11] Math - High | 2 | |
## α’αΆαααααααΆ
ααααα
ααα·ααα·ααααΆααααα·ααααααααααΆααααΈ 11 αα½αααΆααααααΌα ααααα·ααααΆα’αααΈ αααααΈααα·αα’αα»ααΆααα½αααα·ααα·ααααΆ α’αα»ααααα’α·α
αααααΌαααααααααα·αα’αα»ααααααααΆααΈα α’αα»ααααααααΈαααααΆααα αααααααΈααα’αα ααααΌααΆα ααααΈααααααααα»ααααα·α
αα·αααα·α
ααααααα»αααα α αα·ααααααααααΎαααΎαααααα·ααα·ααααΆααααα·ααααααααααΌααααααα·ααα·ααααΆααααα·αααΌαααααΆαααΎααααΈααααααααααα ... | [11] Math - High | 3 | |
# αααααΈα’ααααα
## ααα·ααα·ααααΆααααα·αααααα
| | ααααα |
| :--- | :--- |
| **ααααΌαααΈ 1 : ααααΈααα·αα’αα»ααΆααα½αααα·ααα·ααααΆ** | 1 |
| 1. ααααΌααα½ααααααΈααααααα | 2 |
| 2. ααααΆαααααααα½ααααααΈα | 16 |
| 3. αα·α
αΆαα’αα»ααΆααα½α | 24 |
| **ααααΌαααΈ 2 : α’αα»ααααα’α·α
αααααΌαααααααααα·αα’αα»ααααααααΆααΈα** | 35 |
| 1. α’αα»ααααα’α·α
αααααΌαααααααα | 36 |
| 2. α’... | [11] Math - High | 4 | |
### ααααΌα α‘ αααααααΈ α‘
# ααααΌα 1
# ααααΈααα·αα’αα»ααΆααα½αααα·ααα·ααααΆ
![pyramids.png: Photo of the great pyramids of Giza]
- **ααααΌααα½ααααααΈααααααα**
- **ααααΆαααααααα½αααααααΈα**
- **αα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ**
ααΆααα·ααααΆαααααΈααααααΆαα±ααααα·ααα·ααααΆααΆααααααΆαααΆαααΈαα
ααααΎαα‘αΎαααΈαα½ααααααα
αα½ααααα ααΆααααααααΌααα½αααααααΈαα’αΆα
α±αααααααααΆαααΌαα... | [11] Math - High | 5 | |
## αααααααΈ 1 ααααΌααα½ααααααΈααααααα
### 1. ααααααααΆααααΌα
α
αααααααααΈααααααα αααα»αα’αΆα
ααααΎααΌαααααααααΌααααααααΈαααααααα α¬αααααΈαααααΈααΆαααααααααΆααααΌαααΆααΆαα‘αΎα αααα»αααααΈαααααααΌαααααΎαα·ααΈαααααααα
ααΆααααααααααααααΈα α
#### ααααα»αααα
- ααααΆααααΌααα½αααααααΈα
- α
ααααααΎαα·αα·ααααααααΆ β αααααΆααααααΌααααααααΈα
- ααααααα½ααΈ n ααΆααααααααΈαααααΆαα 1... | [11] Math - High | 6 | |
### ααααΌα α‘ αααααααΈ α‘
$\frac{1}{1 \cdot 2} = \frac{1}{1} - \frac{1}{2}$
$\frac{1}{2 \cdot 3} = \frac{1}{2} - \frac{1}{3}$
$\frac{1}{3 \cdot 4} = \frac{1}{3} - \frac{1}{4}$
...
$\frac{1}{(n-1) \cdot n} = \frac{1}{n-1} - \frac{1}{n}$
$\frac{1}{n \cdot (n+1)} = \frac{1}{n} - \frac{1}{n+1}$
ααΌαα’ααααα·αα’ααα ααααΆα $S = 1 - \... | [11] Math - High | 7 | |
### 2. ααααΆααααΌαααΆαααααΆαααααΌ
α
αααααααααΈααααααΆααα½ααΆα
ααα½αααα ααα’αΆα
ααααΆααααΌαααΆααΆαααΆαααααααααααΆαααααΌ α
$1 = 1 = 1^2$
$1+3 = 4 = 2^2$
$1+3+5 = 9 = 3^2$
$1+3+5+7 = 16 = 4^2$
α ααα»ααα $1+3+5+\dots+(2n-1) = n^2$ α
#### ααα αΆααααααΌ
ααααΆααααΌα n αα½ααα
ααα½αααΌ $S = 2+4+6+\dots+2n$ α
**α
ααααΎα**
$2 = 2 = 1 \times 2$
$2+4 = 6 = 2 \time... | [11] Math - High | 8 | |
### ααααΌα α‘ αααααααΈ α‘
### 3. αα·αα·ααααααααΆ β αααααΆααααααΌααααααααΈα
#### 3.1 αααααΆα β
αααα»αααΆααααααααααΌααα½αααααααΈα $U_1, U_2, U_3, \dots, U_n$ ααααααΎαα·αα·ααααααααΆ β α’αΆαααΆ ααα·α
αααΆ αααααΆααααΆαααααΌα n αα½αααααααΈα α αααααααααααα $\sum_{k=1}^{n} U_k = U_1 + U_2 + U_3 + \dots + U_n$ α αααα»αααα k αααααααααΈ 1, 2, 3, ... αα αΌαααα n ... | [11] Math - High | 9 | |
#### ααα αΆααααααΌ 2
αααααααααααα½ααΆααα’ααααααααΌαααααα·αααααΎαα·αα·ααααααααΆ β
α. $\sum_{k=1}^{6} 2$
α. $\sum_{n=2}^{5} (2n+1)$
α. $\sum_{j=1}^{5} j(j+1)$ α
**α
ααααΎα**
α. $\sum_{k=1}^{6} 2 = 2+2+2+2+2+2$ α
α. $\sum_{n=2}^{5} (2n+1) = 5+7+9+11$ α
α. $\sum_{j=1}^{5} j(j+1) = 1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \... | [11] Math - High | 10 | |
### ααααΌα α‘ αααααααΈ α‘
#### 3.2 ααααΌαααΆαααααααααΌα
ααΆαααααα α
- α. $\sum_{k=1}^{n} c = nc$
- α. $\sum_{k=1}^{n} ca_k = c \sum_{k=1}^{n} a_k$
- α. $\sum_{k=1}^{n} (a_k+b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k$
- α. $\sum_{k=1}^{n} (a_k-b_k) = \sum_{k=1}^{n} a_k - \sum_{k=1}^{n} b_k$
- α. $\sum_{k=1}^{n} (a_k+b_k)^2... | [11] Math - High | 11 | |
#### ααα αΆααααααΌ
ααααΆ
α. $\sum_{k=1}^{15} (4k+3)$
α. $\sum_{k=1}^{20} (k+3)k$ α
**α
ααααΎα**
α. $\sum_{k=1}^{15} (4k+3) = \sum_{k=1}^{15} 4k + \sum_{k=1}^{15} 3 = 4\sum_{k=1}^{15} k + \sum_{k=1}^{15} 3$
$= 4(1+2+3+\dots+15) + 15 \times 3 = 4 \times \frac{(1+15) \times 15}{2} + 45 = 2 \times 16 \times 15 + 45 = 525$ α
α.... | [11] Math - High | 12 | |
### ααααΌα α‘ αααααααΈ α‘
### 4. ααααααααααα½ααΈ n ααΆααααααα½ααααααΈα
ααα’αΆα
ααααααα½ααΈ n αααααααααΈα $(a_n)$ ααααα»ααααααΆαααααΈαααααααα α¬ααΆαααααΈαααααΈααΆαααααΆαααααααΌα
ααΆαααααα α
#### 4.1 αααααα½ααααΆααααΈ 1 ααααααΈα
##### α§ααΆα ααα 1
ααααααα½ααΈ n αααααααΈα $1, 3, 7, 13, 21, 31, \dots$ α
**α
ααααΎα**
ααααααααααΎαααΆαααααΈαααααα·ααααααΆαααααΈαααααα... | [11] Math - High | 13 | |
##### α§ααΆα ααα 2
αααα αΆαααΆ $a_n = a_1 + \sum_{k=1}^{n-1} b_k$ α
αααα $n \ge 2$ α
$a_1 \quad a_2 \quad a_3 \quad a_4 \quad \dots \quad a_{n-1} \quad a_n$
$\quad b_1 \quad b_2 \quad b_3 \quad b_4 \quad \dots \quad b_{n-2} \quad b_{n-1}$
$a_2 - a_1 = b_1$
$a_3 - a_2 = b_2$
$a_4 - a_3 = b_3$
+ ...
$a_{n-1} - a_{n-2} = b_{n-2}... | [11] Math - High | 14 | |
### ααααΌα α‘ αααααααΈ α‘
α
αααα $n \ge 2$ ααααΆα
$a_n = a_1 + \sum_{k=1}^{n-1} b_k = 2 + \sum_{k=1}^{n-1} 2k = 2 + 2 \cdot \frac{1}{2}n(n-1) = 2 + n^2 - n = n^2 - n + 2$
ααα $a_n = n^2 - n + 2$ α
α
αααα $n=1$, $a_1 = 1^2 - 1 + 2 = 2$ αα·α α
ααΌα
ααα αααααΈα $(a_n)$ ααΆααα½ααΈ n αααααααα $a_n = n^2 - n + 2$ α
#### ααα αΆααααααΌ 2
α. ... | [11] Math - High | 15 | |
#### ααααα·ααααα·
1. α. ααααααα½ααΈ n αααααααΈα $4, 5, 7, 10, 14, \dots$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
2. α. ααααααα½ααΈ n αααααααΈα $3, 5, 8, 12, 17, 23, \dots$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
#### 4.2 αααααα½ααααΆααααΈ 2 ααααααΈα
ααΎαα·αααΆαααααΈα $(b_n)$ αα
αααα·αα’αΆα
ααααααα½ααΈ n αααααααΈα $(a_n)$ ααΆαααα αααααααααΌαα... | [11] Math - High | 16 | |
### ααααΌα α‘ αααααααΈ α‘
α
αααα $n \ge 2$ ααααΆα $b_n = b_1 + \sum_{k=1}^{n-1} c_k = b_1 + \sum_{k=1}^{n-1} (2k+1)$
$= 1 + 2\sum_{k=1}^{n-1} k + (n-1) = 1 + 2 \cdot \frac{1}{2}(n-1)n = n^2$ ααα $b_n = n^2$ α
α
αααα $n=1$, $b_1 = 1$ αα·α α ααΌα
ααα $b_n = n^2$ α
α
αααα $n \ge 2$ ααααΆα $a_n = a_1 + \sum_{k=1}^{n-1} b_k = 1 + \sum... | [11] Math - High | 17 | |
αααααΈα $(c_n)$ ααΆαααααΈαααααααααααααΆααα½ααΈ 1 ααααΎααΉα 16 αα·ααααααα½αααααΎααΉα 6
$c_n = 16 + 6(n-1) = 16 + 6n - 6 = 6n + 10$ α
α
αααα $n \ge 2$ ααααΆα $b_n = b_1 + \sum_{k=1}^{n-1} c_k = 14 + \sum_{k=1}^{n-1} (6k+10)$
$= 14 + 6\sum_{k=1}^{n-1} k + 10(n-1) = 14 + 6 \cdot \frac{1}{2}n(n-1) + 10(n-1) = 3n^2 + 7n + 4$ α
α
αααα $n=1$... | [11] Math - High | 18 | |
### ααα αΆαα
#### ααααΌα α‘ αααααααΈ α‘
1. αααααααααΌαααΆαααααααααααααΎαα·αα·ααααααααΆ β :
- α. $1+2+3+\dots+100$
- α. $1+4+9+16+\dots+484$
- α. $1+8+27+64+\dots+3375$
- α. $1 \times 3 + 2 \times 4 + 3 \times 5 + \dots + 20 \times 22$ α
2. αααααααααααα½ααΆααα’ααααααααΌαααααα·αααααΎαα·αα·ααααααααΆ β :
- α. $\sum_{k=1}^{6} k... | [11] Math - High | 19 | |
## αααααααΈ 2 ααααΆαααααααα½ααααααΈα
### 1. ααααααα½ααΈ n αααααααΎαααααΈααααα½α
ααααΆααα·ααααΆαα½α
ααα αΎαααΌαααααΆαααααααα½αααααααΈαααααααα αα·αααααΈαααααΈααΆααα α
α
αααααααααΈαααααααα $a_{n+1} = a_n + d$, d ααΆαααααα½α α
α
αααααααααΈαααααΈααΆααα $a_{n+1} = a_n \times q$, q ααΆααααααα½α α
ααα
αααααααΉααα·ααααΆα’αααΈααααΆαααααααα½αααααααΈαααααααααααααααααααααα½... | [11] Math - High | 20 | |
### ααααΌα α‘ αααααααΈ α’
ααα $a_{n+1} - a_n = 1+4a_n - (1+4a_{n-1}) = 4(a_n - a_{n-1})$ (1)
ααΎααααΆα $b_n = a_{n+1} - a_n$ αααααΈ (1) ααααΆα $b_n = 4b_{n-1} \Rightarrow \frac{b_n}{b_{n-1}} = 4$ ααααααααΈα $(b_n)$ ααΆαααααΈαααααΈααΆααααααααΆαααααααα½αααααΎααΉα 4 αα·αααΆααα½ααΈ 1 αα·ααα½ααΈ n ααΊ
$b_1 = a_2 - a_1 = 1+4 \times 1 - 1 = 4$, $b_n... | [11] Math - High | 21 | |
#### ααα αΆααααααΌ 2
ααα±αααααααΈα $(a_n)$ αααααααα $a_1 = 3$, $a_{n+1} = 2a_n - n + 1$ α ααααααα½ααΈ n α
**α
ααααΎα**
ααΆα $(r_n)$ ααα $r_n = \alpha n + \beta$ ααΆαααΈααααα½α α
ααααΆα $a_{n+1} = 2a_n - n + 1 \Rightarrow \alpha \cdot (n+1) + \beta = 2(\alpha n + \beta) - n + 1$
$n \cdot (\alpha - 1) + \beta - \alpha + 1 = 0$ ααααΆαα... | [11] Math - High | 22 | |
### ααααΌα α‘ αααααααΈ α’
**ααΆααΌαα
**
α
αααα $n \ge 2$ ααααΆα $S_n - S_{n-1} = a_n$ αα·α $S_1 = a_1$ α
**αααααΆαα**
$S_n$ αα·α $S_{n+1}$ ααΆαααΆαα
αΆαααΆα
αααΎααααΈαα $a_{n+1}$ ααΊ $a_{n+1} = S_{n+1} - S_n$ α
#### ααα αΆααααααΌ
ααααΆαααααΌα n αα½ααααΌααααααααΈα $(a_n)$ ααΊ $S_n = n^3 + 2n$ α
ααααααα½ααΈ n αααααααΈα $(a_n)$ α
**α
ααααΎα**
ααααααα½ααΈ... | [11] Math - High | 23 | |
$a_4 - a_3 = 2^3$
...
$a_n - a_{n-1} = 2^{n-1}$
$a_n - a_1 = 2 + 2^2 + 2^3 + \dots + 2^{n-1}$ ααα $a_n = 1 + 2 + 2^2 + 2^3 + \dots + 2^{n-1}$
ααΆααααΌααα½αααααααΈαααααΈααΆααααααααΆα 1 ααΆαα½ααΈαα½α αα·αααααααα½αααααΎααΉα 2
ααααΆα $a_n = 2^n - 1$ α
**ααΆααΌαα
**
ααα’αΆα
ααααΆ $a_n$ ααΆααα·ααΈααΆαααααα :
$a_{n+2} = 3a_{n+1} - 2a_n \Leftrightarrow... | [11] Math - High | 24 | |
### ααααΌα α‘ αααααααΈ α’
ααα’αΆα
αα $\alpha$ αα·α $\beta$ ααααααααααΆααααΈααΆα $x^2 + px + q = 0$ αααααααααααΆααααααΌααα·ααααα»αααΆ α
αααΈααΆα $x^2 + px + q = 0$ α α
ααΆ βαααΈααΆααααααΆαααα $a_{n+2} + pa_{n+1} + qa_n = 0$β α
#### ααα αΆααααααΌ
αααααΈα $(a_n)$ αααααααα $a_1 = 1, a_2 = 13, a_{n+2} = a_{n+1} + 6a_n$ ($n=1, 2, \dots$) α ααααααα½ααΈ ... | [11] Math - High | 25 | |
### ααα αΆαα
1. αααααΈα $(a_n)$ ααααααααααααΆααααααααααΎαααΌα
ααΆαααααα :
- α. $a_1 = 3, a_{n+1} = 2a_n - 4$
- α. $a_1 = 5, a_{n+1} = 3a_n - 2n$ α
ααααααα½ααΈ 4 ααααααΈαααα α
2. αααααΈα $(a_n)$ ααααααααααααΆααααααααααΎα $a_1 = 1, a_2 = 2, a_{n+2} = a_{n+1} + a_n$ α
ααααααα½ααΈ 7 αααααααΈαααα α
3. αααααΈα $(a_n)$ ααααααααααα... | [11] Math - High | 26 | |
### ααααΌα α‘ αααααααΈ α’
8. ααααΆα $S_n$ ααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$ α αΎα $S_n$ ααααααααααααα $S_n = \frac{n}{n-1} a_n$ $n \ge 2$ α
- α. αααααΆαααα $a_n$ ($n \ge 3$) α’αα»ααααααΉα n αα·α $a_{n-1}$ α
- α. αααααΆαααα $S_n$ ($n \ge 2$) α’αα»ααααααΉα n αα·α $S_{n-1}$ α
- α. α§αααΆααΆ $a_1 = 1$ αααα½ααΈ n αααααααΈα $S_n$ ααα $n ... | [11] Math - High | 27 | |
## αααααααΈ 3 αα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
### 1. αααααΆααααααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
ααααααΆααααααααααααΎαα§ααΆα ααααααααααααααΌαα
ααα½ααααααα
ααααΌα 1 αα½ααΆαααα $P(1) = 1 = 1^2$
ααααΌα 2 αα½ααΆαααα $P(2) = 1+3 = 2^2$
ααααΌα 3 αα½ααΆαααα $P(3) = 1+3+5 = 3^2$
ααα’αΆα
ααΆαααΆαααααΌα n αα½ααΆαααα
$P(n) = 1+3+5+\dots+(2n-1) = n^2$ α
#### ααααα»αααα
-... | [11] Math - High | 28 | |
### ααααΌα α‘ αααααααΈα£
#### αααααΆααααααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
**αα·α
αΆαα’αα»ααΆααα½α**
$P(n)$ ααΆααααΎαααααΆααααααΉαα
ααα½αααα n
ααΎααααΈααααΆααααααΆααααΆ $P(n)$ αα·αα
ααααααααα $n \in \mathbb{N}$ ααααααΌα :
1. αααααααααΆααααΆ $P(n)$ αα·αα
αααα $n=1$
2. α§αααΆααΆ $P(n)$ αα·αα
ααααααααα n
3. ααααΆααααααΆααααΆ $P(n)$ αα·αααΆαα±ααααΆα $P(n+1)$ αα·α α
#### ... | [11] Math - High | 29 | |
#### ααα αΆααααααΌ 2
α
αααα $h>0$ αα·αα
ααα½ααααααααααΆαα· $n \ge 2$ αααα αΆαααΆ $(1+h)^n > 1+nh$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
**α
ααααΎα**
α
αααα $n=2$
ααααΆα $(1+h)^2 = 1+2h+h^2 > 1+2h$ ααααα $h^2 \ge 0$
ααΌα
ααα αα·ααααΆαααααα·α α
αααα $n=2$ α
α§αααΆααΆαα·ααααΆαααααα·α α
αααα $n=k$ ($k \ge 2$) ααα $(1+h)^k > 1+kh$ αα·α
ααααΉααααα αΆαααΆααΆαα·ααα... | [11] Math - High | 30 | |
#### ααααα·ααααα·
- α. ααααΆααααααΆαααα·ααααΆα $(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n})(1+2+3+\dots+n) \ge n^2$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
- α. ααααΆααααααΆααααΆ $2^{n+1} + 3^{2n-1}$ ααΆαα α»αα»ααα 7 α
ααααααααα $n \in \mathbb{N}$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
- α. ααααΆααααααΆααααααΆα $1^2+2^2+3^2+\dots+n^2 = \frac{n... | [11] Math - High | 31 | |
- ααααααα»αα
αΆααααααΎααα·αα
α»ααααααααΊ 1 ααΌα
ααααΆ α αΎαααααΌαααααααααα»αααΈααα½ααΆααααααΆαααααααΆα $(x+y)^n$ ααΆααααααα»ααααααααΆα $(x+y)^{n+1}$... α
| | | | | |
| :---: | :---: | :---: | :---: | :---: |
| | 1 | 3 | 3 | 1 |
| 1 | 4 | 6 | 4 | 1 |
ααααααα»ααα $(x+y)^3$
ααααααα»ααα $(x+y)^4$
α¬ααααααα»ααααα½αααααααααααααααΆαα’αΆαααααααΉαααααααα»αα... | [11] Math - High | 32 | |
### ααααΌα α‘ αααααααΈα£
#### ααααα·ααααα·
ααααΎααααΈααααααΆααααΆαααααααΆααααααα $(x+y)^6$ α
ααα
αΆαααΆα
ααααααααααΌαα½ααΌαα
αααααααΆα $(x+y)^n$ ααΊ
$(x+y)^n = 1x^n + ?x^{n-1}y^1 + ?x^{n-2}y^2 + \dots + ?x^{n-r}y^r + \dots + 1y^n$
αα½: 1, 2, 3, ..., (r+1), ..., (n+1)
ααααααα»α: $1, \frac{1 \times n}{1}, \frac{n(n-1)}{1 \times 2}, \dots, \f... | [11] Math - High | 33 | |
#### ααα αΆααααααΌ
αααααααΎαα½ααααααΈ 6 ααααααΈααααααΆααααΆααααααΆααααααα»αααααααααΆ
$C(6,0), C(6,1), C(6,2), C(6,3), C(6,4), C(6,5), C(6,6)$ α
**α
ααααΎα**
```
1 5 10 10 5 1
/ \ / \ / \ / \ / \ / \
1 6 15 20 15 6 1
```
$C(6,0) \ C(6,1) \ C(6,2) \ C(6,3) \ C(6,4) \ C(6,5) \ C(6,6)$
#### ααααα·ααααα·
1. ααα... | [11] Math - High | 34 | |
### ααααΌα α‘ αααααααΈα£
- αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ ααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$, $n \ge 2$ α
- αααααΈα $(c_n)$ ααΆαααααα½ααααΆααααΈ 2 αααααααΈα $(a_n)$ ααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$, $n \ge 2$ α
- α αΎααααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ ααα $b_n = b_1 + \sum_{k=1}^{n-1... | [11] Math - High | 35 | |
### ααα αΆαα
1. ααααΆααααααΆααααΆα
ααααααααα $n \ge 1$; $1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
2. αααα αΆαααΆα
αααααααααα
ααα½αααα n
- α. α
ααα½α $4^n+2$ α
ααααΆα
αααΉα 3 α
- α. α
ααα½α $3^{n+3} - 4^{4n+2}$ α
ααααΆα
αααΉα 11 α
3. ααααΆααααΈα $(U_n)$ ααα n ααΆα
ααα½αααααααααααα $U_{n+1} = \sqrt{U_n... | [11] Math - High | 36 | |
### ααααΌα α‘ αααααααΈα£
### ααα αΆααααααΌα
1. αααααααΈαα½ααααΌαααααααΈαααααααααααααΉαααΆ $S_{10} = 210$ αα·α $S_{20} = 820$ α
2. ααααΉαααΆααααΌααα½ααΈ 1 αα·ααα½ααΈ 4 αααααααΈααααααααααααΎααΉα 2 αα·αααααΌαααΆααααααααΆααααΎααΉα 20 α ααααΆααααΌαααααΆαααΈαα½ααααΌααααααααΈα α
3. ααααΆα $S_m$ αα·α $S_n$ ααΆααααΌα m αα½ααααΌααα·α n αα½ααααΌααααααααΆαααααααΈαααααααααα½αααα... | [11] Math - High | 37 | |
13. αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆααααΆααααααΆααααΆ $\sum_{k=1}^{n} 2^{k-1} = 2^n - 1$ α
14. ααααΆααααααΈα $(U_n)$ ααα n ααΆα
ααα½αααααααααααα $U_{n+1} = 2U_n+1$ αα·α $U_0 = 1$ α αΎααααααΈα $(V_n)$ αααααααα $V_n = U_n+1$ α
- α. αααα αΆαααΆαααααΈα $(V_n)$ ααΆαααααΈαααααΈααΆααα α
- α. ααΆααα $U_n$ ααΆα’αα»αααααα n α
- α. αα·ααααΆααΆαα... | [11] Math - High | 38 | |
### ααααΌα α’ αααααααΈ α‘
# ααααΌα 2
# α’αα»ααααα’α·α
αααααΌαααααααααα·αα’αα»ααααααααΆααΈα
![angkor_wat.png: Photo of Angkor Wat temple complex]
- **α’αα»ααααα’α·α
αααααΌαααααααα**
- **α’αα»ααααααααΆααΈα**
αα
αααα»αααααΌαααα ααα
αΆααααααΎααα·ααααΆααΈαααααααα·ααα’αα»ααααα’α·α
αααααΌαααααααα αααααααΆαααΆαα’αα»αααα αααααααΆααααΈααΆα αα·ααα·αααΈααΆαα’α·α
αααααΌαααααααα αααααΆ... | [11] Math - High | 39 | |
## αααααααΈ 1 α’αα»ααααα’α·α
αααααΌαααααααα
### 1. α’αα»ααααα’α·α
αααααΌαααααααα
#### 1.1 ααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα
αααααααΆαααα’αα»αααααααα»αααααα»ααααα½α
$f(x) = 2^x, g(x) = (1.5)^x$
$h(x) = 1^x, k(x) = (0.5)^x$ α
#### ααααα»αααα
- αααααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα
- αααααααΆααααΈααΆααα·ααα·αααΈααΆαα’α·α
αααααΌαααααααα α
ααΆααΆααααααααααΌααα... | [11] Math - High | 40 | |
### ααααΌα α’ αααααααΈ α‘
- ααΎ $x>0$ ααα $(0.5)^x < 1$ α ααΌα
ααα ααΎ x αααααααααΎαααα‘αΎαα ααα $k(x) = 0.5^x$ ααΆααααααα
α»αααΌα
αα
α α αΎααα·ααα
αα 0 α
- ααΎ $x<0$ ααα $(0.5)^x$ ααΆααααααααα‘αΎααααααΆαααααα α
**ααΆααΌαα
**
α’αα»ααααα’α·α
αααααΌαααααααα $y=a^x$ ααΆαααααΆα :
- ααΎ $a>1$ ααααΆααα $y=a^x$ ααΎαααΈααααααα
ααααΆα ααααΆ $y=a^x$ ααΆα’αα»ααααααΎα
- ααΎ $0... | [11] Math - High | 41 | |
#### 1.2 ααααααααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα
##### α§ααΆα ααα
αααααααΆαααα’αα»αααα
α. $y = 2^x + 1$
α. $y = 2^{x+1}$
α. $y = 2^{x-2}$
α. $y = -2^x$
**α
ααααΎα**
α. αααααααΆαααα’αα»αααα $y = 2^x + 1$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 2^x | y = 2^x + 1 |
|---|---|---|
| -2 | 0.25 | 1.25 |
| -1 | 0.5 | 1.50 |
| 0 | 1 | 2 |
... | [11] Math - High | 42 | |
### ααααΌα α’ αααααααΈ α‘
α. αααααααΆαααα’αα»αααα $y = 2^{x-2}$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 2^x | y = 2^{x-2} |
|---|---|---|
| -2 | 0.25 | 0.06 |
| -1 | 0.5 | 0.12 |
| 0 | 1 | 0.25 |
| 1 | 2 | 0.5 |
| 2 | 4 | 1 |
![graph_y_eq_2_pow_x_minus_2.png: Graph of y=2^x and y=2^(x-2)]
ααΆαααααΆα ααααααααααΎαααΆ ααΎααααΈααααα... | [11] Math - High | 43 | |
**ααΆααΌαα
**
- ααΎααααΈαααααααΆαααα’αα»αααα $y = a^x + q$ ααααααΌααααααααΆα $y = a^x$ αα½α
ααααΎααααααα·αααααα’αααα (oy) α
ααα½α q α―αααΆα‘αΎαααΎααΎ $q>0$ α αΎαα
ααα½α q α―αααΆα
α»ααααααααΎ $q<0$ α
- ααΎααααΈαααααααΆαααα’αα»αααα $y = a^{x-p}$ ααααααΌααααααααΆα $y = a^x$ αα½α
ααααΎααααααα·αααααα’αααα (ox) α
ααα½α p α―αααΆαα
ααΆαααααΆαααΎ $p>0$ α αΎα p α―αααΆαα
ααΆααααααααΎ $p<... | [11] Math - High | 44 | |
### ααααΌα α’ αααααααΈ α‘
#### ααα αΆααααααΌ
αααααααΆααααΈααΆα
- α. $2^{3x+5} = 128$
- α. $5^{x-3} = \frac{1}{25}$
- α. $(\frac{1}{9})^x = 81^{x+4}$
- α. $49^x = 7^{x^2-15}$
- α. $36^{2x} = 216^{x-1}$
- α
. $10^{x-1} = 100^{2x-3}$ α
**α
ααααΎα**
- α. $2^{3x+5} = 128 \Rightarrow 2^{3x+5} = 2^7 \Rightarrow 3x+5=7 \Rightarrow x = \f... | [11] Math - High | 45 | |
$2^{3x+1} < 2^{-5}$ ααααα $\frac{1}{32} = \frac{1}{2^5}$ α¬ $2^{-5}$ α α ααα»ααα α’αααααΆααααΈαααΆααααααΌα
ααααΆ α
$3x+1 < -5$ αααααααααα·αααΈααΆαα’α·α
αααααΌαααααααα
$3x < -6$ αα 1 ααΈα’αααααΆααααΈα ααααΆα $x < -2$ α
ααα’αααααΆααααΈαααΉα 3 α
**αααααααααΆαα**
α±ααααααα x ααΌα
ααΆα -2 ααΌα
ααΆ $x=-3$
$2^{3x+1} < \frac{1}{32}$
$2^{3(-3)+1} < \frac{1}{32}... | [11] Math - High | 46 | |
### ααααΌα α’ αααααααΈ α‘
- α. $(\frac{1}{3})^x + 3(\frac{1}{3})^{x+1} > 12 \Rightarrow [(\frac{1}{3})^x]^2 + (\frac{1}{3})^x - 12 > 0$, ($x \ne 0$) α
ααΆα $t = (\frac{1}{3})^x, t>0$ ααΎαααΆα $t^2+t-12>0$
$t<-4, t>3$
ααα $t>0$ ααΆαα±ααα«ααααααα·αααΈααΆαααΊ $t>3$
α¬ $(\frac{1}{3})^x > 3 = (\frac{1}{3})^{-1}$
ααα $\frac{1}{3... | [11] Math - High | 47 | |
### ααα αΆαα
1. αααααααΆαααα’αα»ααααααΆαααααααααα»αααααα»ααααα½α :
- α. $f(x) = 2^x; g(x) = 5^x; h(x) = 10^x$
- α. $f(x) = (\frac{1}{2})^x; g(x) = (\frac{1}{5})^x; h(x) = (\frac{1}{10})^x$ α
2. α
αΌαααααααα a ααΎααααααααα $f(x) = a^x$ ααΆααααΆαα
ααα»α
ααΈαα½ααααΌα
ααΆαααααα α
- α. A(3, 216)
- α. B(5, 32)
- α. C(3, 512)
- ... | [11] Math - High | 48 | |
### ααααΌα α’ αααααααΈ α‘
6. ααΎ $a>0$ α α
αΌαααααααα a αα·α x αααααααΎα±ααααααΆα αα·ααα·ααααΆαααΆαααααααααααααααΆαα
- α. $a^x = 1$
- α. $a^x > 1$
- α. $0 < a^x < 1$ α
7. αααααααΆαααα’αα»αααα
- α. $f(x) = 2^{|x|}$
- α. $f(x) = x(2^x)$
- α. $f(x) = x^x$ α
8. αααααααΆαααα’αα»ααααααΆαααααα
- α. $y = 2^{x-1}$
- α. $y = 2... | [11] Math - High | 49 | |
## αααααααΈ 2 α’αα»ααααααααΆααΈα
### 1. ααααΆαααα’αα»ααααααααΆααΈα
α’αα»ααααααααΆααΈα ααΆα’αα»ααααα
αααΆαααα’αα»ααααα’α·α
αααααΌαααααααα α α ααα»αααααααΆαααααααΆαααα»αααααΆαααααΉααααααΆαα $y=x$ α
#### ααααα»αααα
- αααααααΆαααα’αα»ααααααααΆααΈα
- αααααααΆααααΈααΆααα·ααα·αααΈααΆαααααΆααΈα α
**α. ααΎααα a > 1**
αααααααΆαααα’αα»αααα $y=4^x$ αα·α $y = \log_4 x$ α
ααΆααΆαααα... | [11] Math - High | 50 | |
### ααααΌα α’ αααααααΈ α’
**α. ααΎααα 0 < a < 1**
αααααααΆαααα’αα»αααα $y = (\frac{1}{4})^x$ αα·α $y = \log_{\frac{1}{4}} x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| y = (1/4)^x | y = log_(1/4) x |
|---|---|---|---|
| x | y | x | y |
| -2 | 16 | 0.25 | 1 |
| -1 | 4 | 0.50 | 0.50 |
| 0 | 1 | 1 | 0 |
| 1 | 0.25 | 2 | -0.50 |
| 2 | 0.06 ... | [11] Math - High | 51 | |
#### ααα αΆααααααΌ
αααααααΆαααα’αα»αααα
α. $y = \log_{10} x$
α. $y = \log_{\frac{1}{10}} x$
**α
ααααΎα**
α. ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = log_10 x |
|---|---|
| 1/100 | -2 |
| 1/10 | -1 |
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
![graph_log_10_x.png: Graph of y=log_10(x)]
ααΎαααααααααΎαααΆ α’αα»αααα $y = \log_{10} x$ ααΆαααααΆαα... | [11] Math - High | 52 | |
### ααααΌα α’ αααααααΈ α’
### 2. ααααααααααΆαααα’αα»ααααααααΆααΈα
αααααααΆαααα’αα»αααα
α. $y = -2 + \log_3 x$
α. $y = \log_3(x-2)$
α. $y = -\log_3 x$ α
**α
ααααΎα**
α. αααααααΆαααα’αα»αααα $y = -2 + \log_3 x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | log_3 x | y = -2 + log_3 x |
|---|---|---|
| 1/3 | -1 | -3 |
| 1 | 0 | -2 |
| 3 | 1 | -1 |... | [11] Math - High | 53 | |
α. αααααααΆαααα’αα»αααα $y = -\log_3 x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = -log_3 x |
|---|---|
| 1/9 | 2 |
| 1/3 | 1 |
| 1 | 0 |
| 3 | -1 |
| 9 | -2 |
![graph_neg_log_3_x.png: Graph of y=log_3(x) and y=-log_3(x)]
ααΎααααΈαααααααΆαααα’αα»αααα $y = -\log_3 x$ ααααΌααααααααααΆαααα’αα»αααα $y = \log_3 x$ αα½α
ααΌαααααΆααααα»αααααααΆ... | [11] Math - High | 54 | |
### ααααΌα α’ αααααααΈ α’
#### ααααα·ααααα·
αααααααΆαααα’αα»αααα
- α. $y = \log_7(x+3)$
- α. $y = \log_7 x + 3$
- α. $y = -\log_7 x$
- α. $y = \log_2(x-1)^2$
- α. $y = 2 - \log_2 x^2$ α
### 3. ααΌαααααααααΌαααα
ααααΉααααΈααααΆαααααα α’αΆα
α±ααα’ααααα·ααααΆα’αα»ααααααααΆααΈα ααααΎαααααΆααααααΌααααααααααΆααΈααα
ααΆ ααααααααα’αΆα
ααααΆααΆα α
ααΎ a, b αα·α x ... | [11] Math - High | 55 | |
- α. $\log_7 27 = \frac{\log_{10} 27}{\log_{10} 7} \approx 1.6937$ α
- α. $\log_5 125 = \log_5 5^3 = 3$ α
#### ααααα·ααααα·
ααααΆ
- α. $\log_5 625$
- α. $\log_5 346$
- α. $\log_6 4870$ α
### 4. αααΈααΆααα·ααα·αααΈααΆαααααΆααΈα
#### 4.1 αααΈααΆαααααΆααΈα
ααΎ $a>0, a \ne 1$ ααααααΈααΆα $\log_a x = \log_a y$ ααααΆα $x=y$ α
##### α§ααΆα ααα... | [11] Math - High | 56 | |
### ααααΌα α’ αααααααΈ α’
$(2\log_9 x - 1)(\log_9 x - 2) = 0$ ααααΌα $\log_9 x = \frac{1}{2}, \log_9 x = 2$
ααΎ $\log_9 x = \frac{1}{2}$ α¬ $x = 9^{\frac{1}{2}} \Rightarrow x=3$
$\log_9 x = 2$ α¬ $x = 9^2 \Rightarrow x=81$ α
#### ααααα·ααααα·
αααααααΆααααΈααΆα
- α. $\log_9 x = \frac{3}{2}$
- α. $\log_x \frac{1}{10} = -3$
- α. $\l... | [11] Math - High | 57 | |
α. $\log_{\frac{1}{3}}(2x-1) \le 2$
$\log_{\frac{1}{3}}(2x-1) \le \log_{\frac{1}{3}}(\frac{1}{3})^2$ ααααα $2 = \log_{\frac{1}{3}}(\frac{1}{3})^2$ ααΆαααααααααααΆααΈαααααα
$2x-1 \ge \frac{1}{9}$ ααααΌααα·ααα
ααααααααααααΎααΉα $\frac{1}{3} < 1$
$x \ge \frac{5}{8}$ α
**αααααααααΆαα**
$\log_{\frac{1}{3}}(2x-1)$ ααΆααααααΎ $2x-1>0$ ... | [11] Math - High | 58 | |
### ααα αΆαα
1. αααααα’αα»ααααα
αααΆαααα’αα»ααααααΆαααααα :
- α. $f(x) = 10^x$
- α. $g(x) = 3^x$
- α. $h(x) = 7^x$
- α. $f(x) = (\frac{1}{2})^x$
- α. $g(x) = (\frac{1}{5})^x$
- α
. $h(x) = (\frac{1}{10})^x$ α
2. αααααα’αα»ααααα
αααΆαααα’αα»ααααααΆαααααα :
- α. $f(x) = \log x$
- α. $g(x) = \log_3 x$
- α. $h(x... | [11] Math - High | 59 | |
7. ααα±ααα’αα»αααα $f(x) = a^x$ αα·αα’αα»ααααα
αααΆα $f^{-1}(x) = \log_a x$ ααα $a>0$ α ααααααα a ααΎααααΈα±αααααααααααα’αα»αααα $f(x)$ αα·α $f^{-1}(x)$ ααΆααααααΆ α
8. ααα±αα $f(x) = x - \log_2 x$ α αΎα $g(x) = 2^x$ α
ααααΆ
- α. $f(g(x))$
- α. $g(f(x))$ α
9. αααααααΆααααΈααΆααα·ααααααααααΆαα
- α. $\log_2(2x+4) - \log_2(x-1) = 3$
... | [11] Math - High | 60 | |
### ααααΌαα£ αααααααΈ α‘
# ααααΌα 3
# αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα
![pendulum_clock.png: Photo of an octagonal pendulum wall clock]
- **αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα**
αα
αααα»αααα·ααα·ααααΆααααα·αααΌαααααΆα ααΎαααΆααα·ααααΆααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆαααααΆαα ααΌα
ααΆ $\cos x = a, \sin x = a, \tan x = t, \cos x > a, \sin x < a, \dots... | [11] Math - High | 61 | |
## αααααααΈ 1 αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα
ααΎαααΆααα·ααααΆαα½α
ααα αΎαααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆαααααΆαα αα
αααα»αααααα
ααα·ααα·ααααΆααααα·αααΌαααααΆα α αααα»αααααααααααΎαααΉααα·ααααΆααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα αααααΆαααααααααααααααα α
#### ααααα»αααα
- αααααααΆααααΈααΆαααααΈαααααΆααα
- αααααααΆααα·αααΈααΆαααααΈαααααΆααα α
### 1. αααΈααΆαααααΈαααα... | [11] Math - High | 62 | |
### ααααΌαα£ αααααααΈ α‘
αααΈααΆα (1) ααΆαα
ααααΎα : $\sin(x+\frac{\pi}{3}) = 0$ ααΆαα±αα $x+\frac{\pi}{3} = k\pi$ α
ααΌα
ααα $x = -\frac{\pi}{3} + k\pi, k \in \mathbb{Z}$ α α
ααα½αα
α»αααααΌα
ααααΎαααΆαααΈα α
**ααααααΈ 2**
ααααΆ $\sin x$ αα·α $\cos x$ ααΆα’αα»αααααα $t = \tan \frac{x}{2}$ ααα $x \ne \pi + 2k\pi, (k \in \mathbb{Z})$ α
αααααα½α $... | [11] Math - High | 63 | |
#### ααααα·ααααα·
αααααααΆααααΈααΆα
- α. $2 \sin x - 3 \cos x = 3$
- α. $\cos 2x - \sin 2x = -1$
- α. $\cos x + \sqrt{3} \sin x = 1$
- α. $\cos x - \sqrt{3} \sin x = 3$ α
#### 1.2 αααΈααΆαααΊααααααΈ 2 αααααΉαα’αα»ααααααααΈαααααΆαααααα
ααα½ααα·α x
αααΈααΆαααΆαααααααΆαααΆα : $a \cos^2 x + b \cos x + c = 0, a \sin^2 x + b \sin x + c = 0$
$a \... | [11] Math - High | 64 | |
### ααααΌαα£ αααααααΈ α‘
ααΌα
ααα $\tan \frac{x}{2} = \tan(-\frac{\pi}{4}), \frac{x}{2} = -\frac{\pi}{4} + k\pi, x = -\frac{\pi}{2} + 2k\pi$
$\tan \frac{x}{2} = \tan \frac{\pi}{3}, \frac{x}{2} = \frac{\pi}{3} + k\pi, x = \frac{2\pi}{3} + 2k\pi, k \in \mathbb{Z}$ α
ααΌα
ααα αααΈααΆαααΆαα
ααααΎα : $x = -\frac{\pi}{2} + 2k\pi$ αα·α $x... | [11] Math - High | 65 | |
### 2. αααΈααΆααααααΆαααΆα $a \sin^2 x + b \sin x \cos x + c \cos^2 x = d$ ααα $a, b, c \ne 0$
ααΎααααΈαααααααΆααααΈααΆαααα ααααααΌαα
ααα’αααααΆααααΈααααααΈααΆαααΉα $\cos^2 x \ne 0, x \ne \frac{\pi}{2} + k\pi$ α
ααααΆααααΈααΆα $a \tan^2 x + b \tan x + c = 0$ αα½α
αααααααΆαααΆααααΈααΆαααΊααααααΈ 2 ααΌα
ααΆαααΎ α
#### α§ααΆα ααα 1
αααααααΆααααΈααΆα $3 \sin^... | [11] Math - High | 66 | |
### ααααΌαα£ αααααααΈ α‘
#### ααα αΆααααααΌ 2
ααααααα m αααααΆαα±αααααΈααΆα $1+m \cos x = m^2 - \cos^2 x$ ααΆαα«α α
**α
ααααΎα**
ααΆα $\cos x = t$ ααα $-1 \le t \le 1$ α
ααααΆα $t^2 + mt + 1 - m^2 = 0$ α ααΆα $f(t) = t^2 + mt + 1 - m^2$
αααΈααΆαααΆαα«ααα½αααΎα
ααααα $[-1, 1]$ ααΆαααΆ $f(-1) \times f(1) \le 0$ α
ααααΆα $(1-m+1-m^2)(1+m+1-m^2) \l... | [11] Math - High | 67 | |
### 3. αααααααααααΈααΆαααααΈαααααΆααα
#### α§ααΆα ααα 1
αααααααΆααααααααααααΈααΆα $\begin{cases} x+y = \frac{\pi}{3} \\ \sin x + \sin y = 1 \end{cases}$
ααααΎααΌααααααααααααΈααααΌααα
ααΆαααα»α ααααΆααααΈααΆα
$\sin x + \sin y = 2 \sin(\frac{x+y}{2}) \cos(\frac{x-y}{2}) = 1$ ααα $x+y = \frac{\pi}{3}$ ααααΆα $2 \sin \frac{\pi}{6} \cos(\frac{... | [11] Math - High | 68 | |
### ααααΌαα£ αααααααΈ α‘
**α
ααααΎα**
ααααΆααααΈααΆα (2) ααααα : $4 \cos(x+y) \cos(y-x) - 1 - 4\cos^2 m = 0$ ααΆα (1)
αααα½αα
αΌα
ααααΆα : $4 \cos^2 m - 4 \cos(x+y) \cos m + 1 = 0, [2 \cos m - \cos(x+y)]^2 + \sin^2(x+y) = 0$
$\begin{cases} \sin(x+y) = 0 \\ \cos(x+y) = 2 \cos m \end{cases} \Rightarrow \begin{cases} x+y = 2k\pi \\ \c... | [11] Math - High | 69 | |
### 4. αα·αααΈααΆαααααΈαααααΆααα
#### 4.1 αα·αααΈααΆααααααΆαααΆα $a \cos x + b \sin x > c$
##### α§ααΆα ααα 1
αααααααΆααα·ααΈαααΆα $\sin x - \cos x > 0$ α
ααααΆα $\sqrt{2}(\frac{\sqrt{2}}{2} \sin x - \frac{\sqrt{2}}{2} \cos x) > 0, \sqrt{2}(\sin x \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos x) > 0, \sqrt{2} \sin(x-\frac{\pi}{4}) > 0$... | [11] Math - High | 70 | |
### ααααΌαα£ αααααααΈ α‘
ααααΆα $0 \le X \le \frac{1}{2}, 0 < X \le \frac{1}{2}, 0 < \sin x \le \frac{1}{2}$ ααααΌαααΉα $2k\pi < x \le \frac{\pi}{6} + 2k\pi$
α¬ $\frac{5\pi}{6} + 2k\pi \le x < \pi + 2k\pi, k \in \mathbb{Z}$ α
α»αααααΌα
ααααΎααα
ααΎααααΌααααΈααΆααα (ααααααα) α
#### ααα αΆααααααΌ 1
αααααααΆααα·αααΈααΆα $2 \cos 2x + \sin^2 x \c... | [11] Math - High | 71 | |
#### ααα αΆααααααΌ 2
αααα αΆαααΆαααα»αααααΈααα ABC ααααΆα $\cos A + \cos B + \cos C \le \frac{3}{2}$
**α
ααααΎα**
ααααΆα $\frac{3}{2} - (\cos A + \cos B + \cos A) = \frac{1}{2}[3-2(\cos A + \cos B + \cos C)]$
$= \frac{1}{2}[1-4 \cos \frac{A+B}{2} \cos \frac{A-B}{2} + 2 - 2 \cos C]$
$= \frac{1}{2}[1-4 \cos \frac{A+B}{2} \cos \frac... | [11] Math - High | 72 | |
### ααααΌαα£ αααααααΈ α‘
## ααααααααααα
ααΎααααΈαααααααΆααααΈααΆα $a \cos x + b \sin x = c$ ααααΆαααΈααααα :
1. ααααααααααΈααΆααααααΆααΆα $\cos(x-\theta) = \frac{c}{r}$ ααα $r = \sqrt{a^2+b^2}, (r \ge 0)$ αα·α $\cos \theta = \frac{a}{r}$, $\sin \theta = \frac{b}{r}$ αα½α
αααααααΆααααΈααΆα $\cos(x-\theta) = \frac{c}{r}$ ααΆαααΌααααα $\cos x... | [11] Math - High | 73 | |
## = ααα αΆαα
1. αααααααΆααααΈααΆαααΆαααααα α
- α. $\cos x + \sqrt{3} \sin x = \cos 3x$
- α. $\sin 3x + 2 \cos x - 2 = 0$
- α. $\sin 2x + \tan x = 2$
- α. $\sin 5x + \cos 5x = \sqrt{2} \cos 13x$
- α. $6 \sin x - 2 \cos^3 x = 5 \sin 2x \cos x$
- α
. $\sqrt{5} \cos x - \cos 2x + 2 \sin x = 0$ α
2. αααααααΆααα... | [11] Math - High | 74 | |
### ααααΌαα£ αααααααΈ α‘
5. αααααααΆααα·αααΈααΆαααΆαααααα α
- α. $\sin^2(\frac{x}{2} - \frac{\pi}{4}) < \cos^2 \frac{2x}{2}$
- α. $6 \sin^2 x - \sin x \cos x - \cos^2 x > 2$
- α. $\frac{\cos x}{1+2 \cos x} > \frac{1-\cos x}{1-2 \cos x}$
- α. $\frac{1-\sin x}{1-3 \sin x} < \frac{1+\sin x}{1-9 \sin^2 x}$ α
6. αααα αΆα... | [11] Math - High | 75 | |
2. αααααααΆααα·αααΈααΆαααΆαααααα α
- α. $2 \cos^2 x - \cos x + 1 \le 0$ αααα»α $[0, \pi]$
- α. $\frac{2 \sin^2 x - \sin x - 1}{\sin x} > 0$ αααα»α $[0, \pi]$
- α. $\frac{\sin x - \cos x + 1}{\sin x + \cos x - 1} > 0$ α
3.
- α. αααααΆααααααα $(x+\frac{1}{x})(x-8)(x-1)$ α
- α. αααααααΆααααΈααΆα $2 \sin^3 2x - 17 \si... | [11] Math - High | 76 | |
### ααααΌα α€ αααααααΈ α‘
# ααααΌα 4 αααααααΈααα’ααα
TAKAKAZU SEKI
## β αααααααΈααα’ααα
α’ααααααααΆαα±ααααΆααααααααα·αααΆααααΈααα»ααα ααΊ Arthur Cayley (1875-1921) α ααΆααα·α αααααααΈαααααΆαααΎαα‘αΎααα»ααααΆααααΈα α αΎαααΆαααααααα
αααα»αα
ααααΎαααΌαα
αααααααααααααΈααΆαααΈααα’ααα α ααΎαααααΎααααααααααΈα... | [11] Math - High | 77 | |
### αααααααΈ 1 αααααααΈααα’ααα
## 1. αααααααΈααα’ααα
### 1.1 αααααΆα
ααΎαααααΆααααααα½α
ααα αΎαααΌααααααα
ααα»α
αα½αα
ααα½ααα
αααα»αααααα α
**α§ααΆα ααα** ααΎαααααααα
ααα»α
$M(x, y)$ αα
ααΆ $M'(x', y')$ ααΆαααααα H αααααΆαααα
α·α O αα·αααααα 4 αααααααΆααααΈααΆα
$x' = 4x$
$y' = 4y$
αααΈααΆααααααααα
αΆααααααααααααααααα α
ααΆαααΈααΆαααΊααααααΈ 1 α¬α α
ααΆαααΈααΆαααΈααα’ααα... | [11] Math - High | 78 | |
### ααααΌα α€ αααααααΈ α‘
ααα’αΆα
ααααΆ $x', y'$ ααΆαααααααα»ααααΆααααΈα $\begin{bmatrix} a & c \\ b & d \end{bmatrix}$ αα·α $\begin{bmatrix} x \\ y \end{bmatrix}$ αα½α
ααααΉααα½αααΆααααΈαααα’αααααΈ 1 αα·ααα½αααΆααααΈαααα’αααααΈ 2 α
**αααααΆαα**
ααΎααααΈααααα½ααααα»αααααα ααα’αΆα
ααααΎα’αααα A ααααααααααααΆαααααααΆααααΆααΆαααΆααααΈααα αα·αααΆαααααααΈααα’ααααα α
... | [11] Math - High | 79 | |
#### ααα αΆααααααΌ 1
αααα αΆαααΆ ααΎαααΆααααΈα A αααααα
ααα»α
$M \ne 0$ αα
ααααΆα
ααα»α
M αααα ααααααΆααααΈα $A = I$ α
**α
ααααΎα**
ααΆα $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααΆαααΆααααΈααααααααα $M(x, y)$ αα
$M(x, y)$ αααα
ααΎαααΆα $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatr... | [11] Math - High | 80 | |
### ααααΌα α€ αααααααΈ α‘
αααααααΆααααααααααααΈααΆα (2) αα·α (4) ααΎαααΆα $b = -1, d = 1$ α
ααΌα
ααα αααΆααααΈααααααααΌαααααΊ $A = \begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix}$ α
#### ααααα·ααααα·
1. αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ ααΆααααΆααααΈα A ααα
$x' = x \cos \alpha - y \sin \alpha, y' = x \sin \alpha + y \cos \alpha$ α ... | [11] Math - High | 81 | |
αααααα
ααα»α
αααααααααααααααααΆα I αα
ααααΆα
ααα»α
αααα½αα―ααααα α
ααααα $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}$ α
#### ααα αΆααααααΌ 1
αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ αααααααα
$\begin{bmatrix} x' \\ y'... | [11] Math - High | 82 | |
### ααααΌα α€ αααααααΈ α‘
#### ααααα·ααααα·
1. αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ αααααααα
- α. $x' = 2y-3, y' = x+1$
- α. $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ -2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} -4 \\ 2 \end{bmatrix}$ α
αααααα
ααα»α
α₯ααααααααα½... | [11] Math - High | 83 | |
ααΌα
ααα $H^{-1} = \begin{bmatrix} \frac{1}{k} & 0 \\ 0 & \frac{1}{k} \end{bmatrix}$ ααΆαααΆααααΈαα
αααΆααα $H = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ α
**ααΆααΌαα
**
ααααααα $M(x, y)$ αα
$M'(x', y')$ ααΆαααααα $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααα
$M' = A(M)$ α¬ $\begin{bmatrix} x' \\ y' \end{bmatrix} = ... | [11] Math - High | 84 | |
### ααααΌα α€ αααααααΈ α‘
ααΎααααΈααααα L' ααααααΌαααα
ααα»α
A', B' αααααΆααΌαααΆααααααααΆααααα
ααα»α
A, B αααααααΆαα L α αΎαααααΆααααΈαα L ααΊααΆαααααΆαααααααΆααααΆαα
ααα»α
A', B' α
αααα $A(0, 1)$ αα·α $B(3, 0)$ ααΆα
ααα»α
αα L αααααααΆα
ααΌαααΆααα A' αααααααα $\begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin... | [11] Math - High | 85 |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 5